Reynolds Number Dependence of Vortex Ring Formation by Transient Jet Ejection

Vortex ring formation by the sudden ejection of a jet from tube and orifice openings is investigated numerically for jet Reynolds number (\textit{Re}) in the range 10 -- 2000 and jet slug length-to-diameter ratios ($L$/$D)$ in the range 0.5 -- 6.0. This \textit{Re} range brackets nearly inviscid behavior (vortex sheet roll-up) at the high end and highly diffusive behavior at the low end. The present investigation is motivated by how the enhanced role of viscosity at low \textit{Re} affects the development and properties of the resulting vortex rings. The results for \textit{Re} = 2000 show classical behavior, namely, compact vortex rings at low $L$/$D$ and a leading vortex ring followed by a trailing jet for $L$/$D$ sufficiently high. As \textit{Re} decreases below 100, viscous diffusion leads to rapid radial growth of the vortex ring trajectories, and rapid decay of total circulation and kinetic energy. For all \textit{Re}, the ratio of the impulse obtained during jet ejection to that from a steady, uniform jet of the same duration increases with $L$/$D$ until a trailing jet appears. The maximum impulse ratio achieved increases as \textit{Re} decreases for the tube configuration, but the opposite trend is observed for the orifice configuration.